Next, we recall the following special case of Lemma 3.1 from [1]}. In the statement below, \(A(n,1)\)  is the set of lines in \(\mathbb {R}^n\)  and if \(Z\subset \ell \) , then \(\mathcal {H}^1_\infty (Z)=\inf \sum |J|\) , where the infimum is taken over all coverings of \(Z\)  by countable collections of open intervals \(\lbrace J\rbrace \) . Note that the analyticity of a set \(M\subset A(1,n)\)  depends on the metric (or more generally, the topology), and the authors in [1]} discuss their choice of metric in Remark 2.2. The metric on \(\mathcal {L}_n\subset A(n,1)\)  described in Section  is compatible with that from [1]}.
